On http://www.simplefit.org/details.html, it says on day 3: "Stay 1-2 repititions below failure, and do not train to failure". Let's say that I'm on level 1 my 'failure point' is right after I finished the 7th pullup. Does this mean that on day 3 I could do 5 pull-ups, stop to rest for - say - one minute (without stopping the stopwatch), and then do the 5 remaining ones? Or should I try an easier variation of the exercise in order to complete all 10 without 'stopping the set' (let's consider that the set is stopped as soon as I put my feet on the floor)?

I believe something along the lines of your first suggestion would be best; however, you do not want long rest periods.

For example, I'm on Level 2, and I do 9-10 pull ups then wait 5 seconds do 1 pull up, etc until 13. Same thing for push ups, last time I did 10,5,5,3,2,1 Whichever way you find to groups up sets is fine, just try not to treat it like Day 2. You should be going all out on Day 3.

If you can do several repetitions of a certain exercise then focus on using those as opposed to just finding an easier variation.